SOLUTION: please. could you explain in details for this college algebra problem? The population of a certain endangered species of owl is declining exponentially. There are currently

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Question 424440: please. could you explain in details for this college algebra problem?


The population of a certain endangered species of owl is declining exponentially. There are currently 450 living specimens, whereas just 8 years ago there were 12,000 . If the population continues to decline exponentially, how long will it be until there is only a single owl left?
Found 2 solutions by nerdybill, Gersid:
Answer by nerdybill(7384) About Me  (Show Source):
You can put this solution on YOUR website!
The population of a certain endangered species of owl is declining exponentially. There are currently 450 living specimens, whereas just 8 years ago there were 12,000 . If the population continues to decline exponentially, how long will it be until there is only a single owl left?
.
apply exponential equation:
y = xe^(kt)
where
y is amount after time t
x is initial amount
k is a constant
t is time
.
plug in all given data to find k:
450 = 12000e^(8k)
450/12000 = e^(8k)
ln(450/12000) = 8k
ln(450/12000)/8 = k
-0.41043 = k
.
Our general equation is:
y = 12000e^(-0.41043t)
.
we now set y to 1 and solve for t:
1 = 12000e^(-0.41043t)
1/12000 = e^(-0.41043t)
ln(1/12000) = -0.41043t
ln(1/12000)/(-0.41043) = t
22.89 years = t




Answer by Gersid(33) About Me  (Show Source):
You can put this solution on YOUR website!
This is a problem in exponential decay for which you can use the exponential function: y+=+ab%5Ex, where the base, b, is between 0 and 1 for exponential decay. x is the number of years given in the problem (x = 8).
Start with the exponential function:
y+=+ab%5Ex Substitute y = 450, the current number of owls. a = 12,000, the number of owls 8 years ago, and x = 8.
450+=+%2812000%29b%5E8 Solving for b will give you the exponential function for this situation.
b%5E8+=+450%2F12000 Use your calculator to take the eighth root of both sides.
b+=+0.6634 Approx. Now you can write the exponential function for this situation:
y+=+12000%280.6634%29%5Ex
The question you want to answer is - For what value of x (x = number of years) will y (the number of owls remaining) = 1.
1+=+12000%280.6634%29%5Ex Solve for x. Divide both sides by 12000.
1%2F12000+=+%280.6634%29%5Ex Take the logarithm of both sides.
Log%281%2F12000%29+=+xLog%280.6634%29 From the power rule for logarithms.
x+=+Log%281%2F12000%29%2FLog%280.6634%29 Use your calculator to evaluate.
x+=+22.8878 years.